Introduction: In this lesson, three trigonometric ratios (sine, cosine, and tangent) will be defined and applied. These involve ratios of the lengths of the sides in a right triangle.

In a right triangle, one angle is 90º and the side across from this angle is called the hypotenuse. The two sides which form the 90º angle are called the legs of the right triangle. We show a right triangle below. The legs are defined as either “opposite” or “adjacent” (next to) the angle A.

We can also find the measure of the angle θ when we know any of these three trigonometric ratios. In this case, tan(θ) = . Using a calculator, we can determine what angle has this tangent. Using a TI-83 calculator, we press and get

Let's Practice:

In the triangle below, what is the sin(θ) and what is the length of the adjacent side? What is the measure of angle θ?

The adjacent side can be found using the Pythagorean Theorem and is . The sin(θ) = Using we get . Notice that the adjacent side is . If we had used we would still get .